The stationary solution map $X$ of a canonically perturbed nonlinear program or variational condition is studied. The focus is on characterizations for $X$ to be locally single-valued and Lipschitz near some stationary point $x^0$ of an initial problem, where the Constraint Qualification MFCQ is satisfied. Since such conditions involve a non-singularity property of the strict graphical derivative $TX$ of $X$, explicit formulas for $TX$ are presented. It turns out that - even for polynomial convex problems - our stability does not only depend on certain derivatives of the problem functions at $x^0$. This is in contrast to various other stability concepts and holds in a similar way for the also characterized Aubin property of the same mapping. Further, we add essential examples, clarify completely the relations to Kojima's strong stability and present simplifications for linearly constrained problems, nonlinear convex programs and for the map of global minimizers as well.